Theory of Valuations on Manifolds: A Survey

Alesker, Semyon

doi:10.1007/s00039-007-0631-x

Cited by 74 publications

(99 citation statements)

References 40 publications

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“…However we note that Alesker [5][6][7][8]10] has introduced a much broader notion of valuation that makes sense even on smooth manifolds. The restricted class of valuations considered here is in some sense an infinitesimal version of this broader notion.…”

Section: Valuationsmentioning

confidence: 98%

Convolution of convex valuations

Bernig

2006

Geom Dedicata

130

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We show that the natural "convolution" on the space of smooth, even, translation-invariant convex valuations on a euclidean space V, obtained by intertwining the product and the duality transform of S. Alesker J. may be expressed in terms of Minkowski sum. Furthermore the resulting product extends naturally to odd valuations as well. Based on this technical result we give an application to integral geometry, generalizing Hadwiger's additive kinematic formula for SO(V) Convex Geometry, North Holland, 1993 to general compact groups G ⊂ O(V) acting transitively on the sphere: it turns out that these formulas are in a natural sense dual to the usual (intersection) kinematic formulas.

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Section: Valuationsmentioning

confidence: 98%

Convolution of convex valuations

Bernig

2006

Geom Dedicata

130

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“…The space of normal densities on R n can be identified with a subspace of translation invariant valuations on compact convex sets. Under this identification, the product of smooth normal densities coincides with Alesker product of smooth valuations, see [5]. Therefore (1.3) can be regarded as an identity from the valuation theory, see 3.3.…”

Section: Introductionmentioning

confidence: 95%

Average number of zeros and mixed symplectic volume of Finsler sets

Akhiezer¹,

Kazarnovskii²

2018

Geom. Funct. Anal.

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Let X be an n-dimensional manifold and V 1 , . . . , V n ⊂ C ∞ (X, R) finite-dimensional vector spaces with Euclidean metric. We assign to each V i a Finsler ellipsoid, i.e., a family of ellipsoids in the fibers of the cotangent bundle to X. We prove that the average number of isolated common zeros of f 1 ∈ V 1 , . . . , f n ∈ V n is equal to the mixed symplectic volume of these Finsler ellipsoids. If X is a homogeneous space of a compact Lie group and all vector spaces V i together with their Euclidean metrics are invariant, then the average numbers of zeros satisfy the inequalities, similar to Hodge inequalities for intersection numbers of divisors on a projective variety. This is applied to the eigenspaces of Laplace operator of an invariant Riemannian metric. The proofs are based on a construction of the ring of normal densities on X, an analogue of the ring of differential forms. In particular, this construction is used to carry over the Crofton formula to the product of spheres.

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“…This is a consequence of Alesker's irreducibility theorem [2] and the kernel theorem of [12]. Here it is important that SU (n) acts transitively on the unit sphere, compare [7] and [18] for more information.…”

Section: Space Alesker Has Shown That Valmentioning

confidence: 99%

“…In fact, this product can even be extended to the much larger space of smooth valuations on a smooth manifold. We refer to [4] and [7] for the definition and the properties of this product.…”

Section: Classification Of Invariant Valuations Of Weight 1 If N Is Oddmentioning

confidence: 99%